Question 7: Write the following as a single logartihm:
\((iii)\) \(\frac{1}{3}( log_{5} \ 8+log_{5} \ 27) -log_{5} \ 3\)
Solution:
\[
\begin{align*}
&=\frac{1}{3}(\log_{5} 8+\log_{5} 27) -\log_{5} 3\\
&=\frac{1}{3}(\log_{5}( 8\times 27)) -\log_{5} 3\\
&=\frac{1}{3}(\log_{5}( 216)) -\log_{5} 3\\
&=\left(\log_{5}\left( 6^{3}\right)^{\frac{1}{3}}\right) -\log_{5} 3\\
&=\log_{5} 6-\log_{5} 3\\
&=\log_{5}\left(\frac{6}{3}\right)\\
&=\log_{5} 2\\
\end{align*}
\]